If f(x) approaches only one value as x ap proaches a, the value approached is said to be the limit of f(x) (or limiting value of f(x)) as x approaches a, and is indicated by the sym bol L f(x) or simply Lf(x). If Lf(x)=f(a) x=a the function f(x) is said to be continuous at the point x = a. A necessary and sufficient condi tion for the existence of the equality Lf(x)=b is the following: for every e > 0 there exists a de > 0, such that for every x+a for which Ix — al,
The proof of these necessary and sufficient conditions and also of the following theorems must be omitted for lack of space. If fi(x) and fe(x) approach definite limiting values as x ap proaches a, then fi(x) + fe(x), and b(z). j.(x) approach Lfs(x) Lfa(x), 111(x)— Lfa(x), and Lf.(x).Lfs(x), respec tively, as x approaches a; if Lfe(x)+0, then fi(x)/fe(x) approaches Lf1(x)/Lh(x). These propositions are evidently to be thought of also as propositions about continuous functions. To them should be added the statement that if f,(x) is continuous at x=a and fa(y) is con tinuous at yf(a), then fe(f.(x)) is continu ous at x—a. Also, if f(x) is continuous at a point .s a, and if f(a) is positive, then there is a neighborhood of x--a upon which the func tion is positive.' In the above propositions about limits it is not necessary that f(x) be defined at every point of an interval. Everything we have said about Lf (x) remains valid when x takes values on any set of which a is a limit point. Indeed, in one of the most important cases, that of the summation of series (q,v.), the independent variable takes on only integral values, and a + co. For the following theorems, how ever, we assume that f(x) is defined at every point of an interval.
Continuous is continuous on an interval ab if it is continuous at every point of the interval; in particular, it must be continuous at a and b. One of the most famous propositions of real function theory states the principle of uniform continuity:* if a is any positive number, there is ad, > 0, such that on every interval of ab of length less than de the oscillation of f(x) is less than e. Other propo sitions in this connection are that a continuous function has a finite maximum value f(x.) and
a finite minimum value f(s,) and takes on all intermediate values.' An important corollary of this is that every algebraic equation of odd degree with real coefficients has one real root. If a function is monotonic and continuous, it has a monotonic and continuous inverse.
If Mx) and fa (x) are continuous on an interval ab and if fa(x) Mx) on a set every where dense on ab, then fi(x)= fa(x) at every point of the interval ab. The last theorem en ables us to say that if a continuous function is known at every point of an everywhere-dense subset of an interval, it is known on the whole interval. Since there are enumerable every where-dense subsets, a continuous function is thus capable of determination by an enumerable set of conditions. This phenomenon is also manifested in the theorem that any continuous function may be expressed as a convergent series of polynomials. Generalizations of this theorem to discontinuous functions have been obtained by R. Baire," the next class of func tions beyond the continuous being those discon tinuous functions which are representable by series of continuous functions.
The class of continuous functions includes as a subclass the differentiable functions, i.e.. those continuous functions which possess at every point a derivative. (See CaLcut.urs). The existence of a derivative, which amounts to the condition that the graph of f(x) shall possess a tangent, puts a limitation on the manner in which a function may oscillate.' The earlier students of calculus assumed that every con tinuous function may be differentiated. in fact. several proofs of this statement were offered. Weierstrass' was the first to give an example of a function which is everywhere continuous and such that for no interval is it differentiable at every point A subclass of the differentiable functions is constituted by those functions for which the second, third and in general the nth derivatives exist. For such a function a formal expansion by means of Taylor's formula (see SERIES; CAL CULUS) is always possible, but as has been shown by A. Cauchy and A. Pringsheim," this series need not always converge and, if con vergent, need not represent function. A function which may be represented by Taylor's series is called analytic. Necessary and suffi cient conditions for the expansibility of func tions by Taylor's series have been given by Pringsheim, but the full significance of the ex istence of all the derivates of a continuous func tion is still to be determined.